How do you evaluate an expression with multiple variables?

To evaluate an expression with multiple variables, you substitute each variable with its given value and then perform the arithmetic operations. For example, consider the expression 1 2 a + 4 b where a = 10 and b = -6 . Replace a and b with these values to get 1 2 × 10 + 4 × - 6 . Calculate each term: 1 2 × 10 = 5 and 4 × - 6 = - 24 . Then add: 5 + - 24 = - 19 . This method applies regardless of how many variables are present.

1. Review of Real Numbers

Evaluating Expressions

1. Review of Real Numbers

Evaluating Expressions: Videos & Practice Problems

Video Lessons

Topic summary

Algebraic expressions combine numbers, variables, and operations, where variables like $x$ , $y$ , or $z$ represent changing values. Coefficients are constant numerical factors multiplying variables, while constants are fixed numbers without variables. Evaluating expressions involves substituting variables with given values and applying operations, including exponents, as in $y^{3}$ . Understanding terms, coefficients, and exponents is essential for working with polynomials, monomials, and multivariable polynomials, laying the foundation for mastering concepts like degree, standard form, and scientific notation.

concept

Evaluating Expressions

Video duration:

Evaluating Expressions Video Summary

Algebraic expressions are mathematical phrases that combine numbers and variables using operations such as addition, subtraction, multiplication, and division. A variable is a letter that represents an unknown or changeable value, commonly denoted by letters like x, y, or z. Unlike variables, a coefficient is a fixed number multiplying the variable, and a constant is a standalone number without any variable attached. Both coefficients and constants remain unchanged, while the value of a variable can vary.

To work with algebraic expressions effectively, one key skill is to evaluate them by substituting variables with specific numerical values. For example, given the expression \$2x + 5$, if $x = 4$, substituting yields $2 \times 4 + 5$. Applying the order of operations, multiplication is performed first, resulting in $8 + 5 = 13$. Thus, the expression evaluates to 13 for $x = 4\(.

When expressions contain multiple variables, such as \)\frac{1}{2}a + 4b$, each variable is replaced with its corresponding value. For instance, if $a = 10$ and $b = -6$, the expression becomes $\frac{1}{2} \times 10 + 4 \times (-6)$. Calculating each term gives $5 + (-24)$, which simplifies to $-19\(. This demonstrates how to handle multiple variables during evaluation.

Exponents also play a role in algebraic expressions. Consider the expression \)-8y^3$ with $y = 2$. Substituting the variable results in $-8 \times 2^3$. Since $2^3$ means $2 \times 2 \times 2 = 8$, the expression simplifies to $-8 \times 8 = -64$. Understanding how to evaluate powers is essential for correctly simplifying expressions involving exponents.

Mastering algebraic expressions involves recognizing the roles of variables, coefficients, and constants, and applying substitution and order of operations to evaluate expressions accurately. This foundational knowledge is crucial for progressing in algebra and solving more complex mathematical problems.

Problem

Evaluate the following algebraic expression for the given value of $x$ .
$x^{2} + 4 x + 6; x = - 2$

$6$

$2$

$negative 2$

$12$

Problem

Evaluate the following algebraic expression for the given value of $x$ .
$7 x^{3} + 2 x - 9; x = 0$

$9$

$negative 2$

$0$

$negative 9$

example

Evaluating Expressions Example 1

Video duration:

Evaluating Expressions Example 1 Video Summary

To evaluate the algebraic expression 2a² + 5b given specific values for a and b, substitute the provided numbers directly into the expression. For example, if a = -3 and b = 4, replace every instance of a with -3 and every instance of b with 4. This results in the expression:

\$2 \times (-3)^2 + 5 \times 4\(

First, calculate the square of -3, which is 9, since squaring a negative number yields a positive result. Then multiply by 2:

\)2 \times 9 = 18\(

Next, multiply 5 by 4:

\)5 \times 4 = 20\(

Finally, add these two results together to find the value of the expression:

\)18 + 20 = 38$

This process demonstrates how to evaluate algebraic expressions by substituting variables with given values and performing arithmetic operations in the correct order. Understanding this method is essential for solving a wide range of problems involving algebraic expressions.

example

Evaluating Expressions Example 2

Video duration:

Evaluating Expressions Example 2 Video Summary

To evaluate the expression a − 3b² for given values of a and b, substitute the provided numbers directly into the formula. For example, if a = −5 and b = 2, replace a with −5 and b with 2, resulting in the expression:

$-5 - 3 \times 2^2$

Next, calculate the exponent first, as per the order of operations. Squaring 2 gives:

\$2^2 = 4\(

Then multiply this result by 3:

\)3 \times 4 = 12\(

Now, subtract this product from −5:

\)-5 - 12 = -17$

Thus, the value of the expression a − 3b² when a = −5 and b = 2 is −17. This process highlights the importance of correctly applying substitution and following the order of operations, especially handling exponents before multiplication and subtraction.

example

Evaluating Expressions Example 3

Video duration:

Evaluating Expressions Example 3 Video Summary

When evaluating algebraic expressions with given values for variables, it is essential to substitute the values accurately and simplify step-by-step. For example, if x equals negative three halves and y equals six, consider the expression 4x + 2/3. Substituting x = -\frac{3}{2} transforms the expression into 4 \times -\frac{3}{2} + \frac{2}{3}. Multiplying fractions involves multiplying numerators and denominators directly, so 4 \times -\frac{3}{2} becomes \frac{4}{1} \times \frac{-3}{2} = \frac{-12}{2}, which simplifies to -6. Adding the remaining fraction yields -6 + \frac{2}{3}. Combining these terms requires expressing the integer as a fraction with a common denominator: -6 = -\frac{18}{3}, so the sum is -\frac{18}{3} + \frac{2}{3} = -\frac{16}{3}. This fraction cannot be simplified further, so the final value of the expression is -\frac{16}{3}.

For a more complex expression such as 2x - 0.5y + 20, substituting x = -\frac{3}{2} and y = 6 gives 2 \times -\frac{3}{2} - 0.5 \times 6 + 20. Multiplying 2 \times -\frac{3}{2} simplifies directly to -3 because the 2 in the numerator and denominator cancel out. The term 0.5 \times 6 equals 3, so the expression becomes -3 - 3 + 20. Combining these values, -3 - 3 = -6, and adding 20 results in 14. Therefore, the evaluated expression equals 14.

Understanding how to substitute values and simplify expressions involving fractions and decimals is crucial in algebra. Multiplying fractions requires multiplying numerators and denominators separately, while simplifying fractions involves dividing numerator and denominator by their greatest common divisor. Additionally, converting decimals like 0.5 to fractions (e.g., 0.5 = \frac{1}{2}) can make calculations more intuitive. Mastery of these skills enables accurate evaluation of algebraic expressions and supports further problem-solving in mathematics.

example

Evaluating Expressions Example 4

Video duration:

Evaluating Expressions Example 4 Video Summary

To evaluate the algebraic expression 2x + 3y - z² when given specific values for the variables, substitute each variable with its corresponding value. For example, if x = 1, y = 4, and z = -2, replace each variable in the expression accordingly. This results in:

\$2 \times 1 + 3 \times 4 - (-2)^2\(

Next, perform the multiplication operations:

\)2 + 12 - (-2)^2\(

Since squaring a negative number results in a positive value, calculate \)(-2)^2\( as:

\)(-2) \times (-2) = 4\(

Substitute this back into the expression:

\)2 + 12 - 4\(

Adding and subtracting in order, first add 2 and 12:

\)14 - 4\(

Then subtract 4 from 14 to get the final result:

\)10$

This process demonstrates how to evaluate expressions with multiple variables by systematically substituting values and applying arithmetic operations, reinforcing the understanding of algebraic evaluation and the importance of correctly handling exponents and negative numbers.

Here’s what students ask on this topic:

Evaluating an algebraic expression means substituting the variables in the expression with given numerical values and then performing the arithmetic operations to find the expression's value. For example, if you have the expression $2x + 5$ and you are asked to evaluate it for $x = 4$ , you replace $x$ with 4, resulting in $2 \times 4 + 5$ . Then, perform the multiplication and addition to get $8 + 5 = 13$ . This process helps you find the numerical value of the expression for specific variable values.

In algebraic expressions, coefficients and constants serve different roles. A coefficient is a fixed number that multiplies a variable. For example, in $2x$ , 2 is the coefficient of the variable $x$ . Coefficients remain constant and do not change. On the other hand, a constant is a number without any variable attached, such as the 5 in $2x + 5$ . Constants also remain fixed and do not vary. Variables, unlike coefficients and constants, represent values that can change. Understanding these distinctions is essential for correctly interpreting and evaluating algebraic expressions.

To evaluate an expression with multiple variables, you substitute each variable with its given value and then perform the arithmetic operations. For example, consider the expression $\frac{1}{2} a + 4 b$ where $a = 10$ and $b = -6$ . Replace $a$ and $b$ with these values to get $\frac{1}{2} \times 10 + 4 \times - 6$ . Calculate each term: $\frac{1}{2} \times 10 = 5$ and $4 \times - 6 = - 24$ . Then add: $5 + - 24 = - 19$ . This method applies regardless of how many variables are present.

Exponents indicate repeated multiplication of a base number or variable. When evaluating algebraic expressions with exponents, you first calculate the power before performing other operations. For example, in the expression $- 8 y^{3}$ with $y = 2$ , you calculate $2^{3}$ which is $2 \times 2 \times 2 = 8$ . Then multiply by the coefficient: $- 8 \times 8 = - 64$ . Understanding the order of operations, especially handling exponents first, is crucial for accurate evaluation.

Variables are fundamental in algebraic expressions because they represent unknown or changing values. Unlike constants and coefficients, variables can take on different values, allowing expressions to model a wide range of situations. For example, in $2x + 5$ , $x$ can be any number, and the expression's value changes accordingly. Understanding variables helps you evaluate expressions for different inputs, solve equations, and analyze functions. This flexibility is essential for applying algebra to real-world problems and higher-level math concepts.